Problem 6 - Precalculus: Mathematics for Calculus

Question

The formula \(d=r t\) models the distance \(d\) traveled by an object moving at the constant rate \(r\) in time \(t .\) Find formulas for the following quantities. \(r=\)_____ \(t=\) _____

Step-by-Step Solution

Answer

The formulas are \( r = \frac{d}{t} \) and \( t = \frac{d}{r} \).

Step 1: Understanding the given formula

The given formula is \( d = rt \), which models the distance \( d \) traveled by an object moving at a constant rate \( r \) over a certain time \( t \). This formula expresses distance as a product of rate and time.

Step 2: Re-arrange to solve for rate \( r \)

To find the formula for the rate \( r \), we need to isolate \( r \) in the equation \( d = rt \). We can do this by dividing both sides of the equation by \( t \). Thus, the formula for the rate \( r \) is:\[ r = \frac{d}{t} \]

Step 3: Re-arrange to solve for time \( t \)

To find the formula for time \( t \), we need to isolate \( t \) in the equation \( d = rt \). We can do this by dividing both sides of the equation by \( r \). Thus, the formula for time \( t \) is:\[ t = \frac{d}{r} \]

Key Concepts

Rate Calculation

The rate of an object refers to how fast it travels over a certain period of time. Understanding how to calculate the rate is crucial in solving distance problems. From the original formula \(d = rt\), if we want to find the rate \(r\), we need to rearrange the formula so that \(r\) stands alone. By dividing both sides of the equation by \(t\), we can isolate \(r\) on one side. This gives us the formula \(r = \frac{d}{t}\).

This tells us that the rate is equal to the total distance traveled divided by the time it took to travel that distance. Remember:

• Make sure your distance \(d\) and time \(t\) are in compatible units (e.g., kilometers and hours).
• The calculated rate \(r\) will share these units, such as kilometers per hour (km/h).

Calculating the rate helps in determining how fast an object is moving, which can be useful in various real-life scenarios, like calculating travel speed for trips.

Time Calculation

Time calculation involves determining how long it takes for an object to cover a certain distance at a constant rate. Using the original formula \(d = rt\), you can find the time \(t\) by rearranging this equation to solve for \(t\).

To do this, divide both sides of the equation by \(r\) to isolate the variable \(t\). This results in the formula \(t = \frac{d}{r}\). Here:

• Ensure that the distance \(d\) and rate \(r\) are in compatible units.
• The result for time \(t\) will be in time units such as hours or seconds, depending on the rate’s unit.

Knowing how to find the time is essential, especially when planning activities like traveling or timing events, as it helps in estimating how long a particular journey or action will take.

Rearranging Equations

Rearranging equations is a fundamental skill in mathematics that allows you to solve for a specific variable in an equation. In the context of the distance formula \(d = rt\), rearranging helps us find either the rate \(r\) or the time \(t\) based on the known values of distance \(d\) and either the other variable.

• To solve for \(r\), divide both sides by \(t\). This gives \(r = \frac{d}{t}\).
• To solve for \(t\), divide both sides by \(r\), resulting in \(t = \frac{d}{r}\).
• Remember that the operations done to one side of the equation must also be done to the other to maintain equality.

Understanding how to rearrange equations expands your ability to tackle various mathematical and real-life problems efficiently and accurately. This skill not only helps in algebra but also in skills such as problem-solving and logical thinking.